A potter makes a bowl by turning the curve x² = 8y about the y-axis…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
A potter makes a bowl by turning the curve about the -axis. The and are in centimetres, the lowest point of the bowl is the origin, and the bowl is cm deep. A tap pours water into the empty bowl at cm per second. The potter wants three numbers. The first is the volume the bowl holds when full. The second is the depth of water when the bowl holds exactly cm. The third is the time the tap needs to fill the bowl to the top. Careful: the water surface is a circle, and its radius changes with the depth, so the volume is not the depth times a fixed area. Ignore the thickness of the clay, and treat the curve as the inside of the bowl. Give the volumes in cm, the depth in cm to two decimal places, and the time in seconds to two decimal places.
Need a reminder? The key ideaShow
For a region turned about the -axis, a slice is a disc of volume , so . With a hole, use washers: . About the -axis, use and write in terms of .
The worked solution
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Given. The bowl is for , turned about the -axis. The tap gives cm per second.
Step 1. Volume up to depth . .
Step 2. The full bowl. At , cm.
Step 3. The depth for cm. gives , so cm.
Step 4. The time. s.
Answer: cm, a depth of cm, and s.
Check. The depth cm is less than cm, as it must be, because cm is less than the full volume.
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